Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYFinally we note that, by analogy with the single-variable case, the characteristicfunction and the cumulant generating function of a multivariate distribution aredefined respectively asC(t 1 ,t 2 ,...,t n )=M(it 1 ,it 2 ,...,it n ) and K(t 1 ,t 2 ,...,t n )=lnM(t 1 ,t 2 ,...,t n ).◮Suppose that the random variables X i , i =1, 2,...,n,aredescribedbythePDFf(x) =f(x 1 ,x 2 ,...,x n )=N exp(− 1 2 xT Ax),where the column vector x =(x 1 x 2 ··· x n ) T , A is an n × n symmetric matrix and Nis a normalisation constant such that∫∫ ∞ ∫ ∞ ∫ ∞f(x) d n x ≡ ··· f(x 1 ,x 2 ,...,x n ) dx 1 dx 2 ···dx n =1.∞Find the MGF of f(x).−∞−∞−∞From (30.142), the MGF is given by∫M(t 1 ,t 2 ,...,t n )=N exp(− 1 2 xT Ax + t T x) d n x, (30.144)∞where the column vector t =(t 1 t 2 ··· t n ) T . In order to evaluate this multiple integral,we begin by noting thatx T Ax − 2t T x =(x − A −1 t) T A(x − A −1 t) − t T A −1 t,which is the matrix equivalent of ‘completing the square’. Using this expression in (30.144)and making the substitution y = x − A −1 t,weobtainM(t 1 ,t 2 ,...,t n )=c exp( 1 2 tT A −1 t), (30.145)where the constant c is given by∫c = N exp(− 1 2 yT Ay) d n y.∞From the normalisation condition for N, weseethatc = 1, as indeed it must be in orderthat M(0, 0,...,0) = 1. ◭30.14 Transformation of variables in joint distributionsSuppose the random variables X i , i =1, 2,...,n, are described by the multivariatePDF f(x 1 ,x 2 ...,x n ). If we wish to consider random variables Y j , j =1, 2,...,m,related to the X i by Y j = Y j (X 1 ,X 2 ,...,X m ) then we may calculate g(y 1 ,y 2 ,...,y m ),the PDF for the Y j , in a similar way to that in the univariate case by demandingthat|f(x 1 ,x 2 ...,x n ) dx 1 dx 2 ···dx n | = |g(y 1 ,y 2 ,...,y m ) dy 1 dy 2 ···dy m |.From the discussion of changing the variables in multiple integrals given inchapter 6 it follows that, in the special case where n = m,g(y 1 ,y 2 ,...,y m )=f(x 1 ,x 2 ...,x n )|J|,1206